Data.Colour.CIE:cieLAB from colour-2.3.3, A

Average Error: 0.2 → 0.2

Time: 5.7s

Precision: binary64

Math

\[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]

↓

\[y \cdot \left(3 \cdot x\right) + y \cdot -0.41379310344827586 \]

FPCore

(FPCore (x y) :precision binary64 (* (* (- x (/ 16.0 116.0)) 3.0) y))

↓

(FPCore (x y)
 :precision binary64
 (+ (* y (* 3.0 x)) (* y -0.41379310344827586)))

C

double code(double x, double y) {
	return ((x - (16.0 / 116.0)) * 3.0) * y;
}

↓

double code(double x, double y) {
	return (y * (3.0 * x)) + (y * -0.41379310344827586);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x - (16.0d0 / 116.0d0)) * 3.0d0) * y
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y * (3.0d0 * x)) + (y * (-0.41379310344827586d0))
end function

Java

public static double code(double x, double y) {
	return ((x - (16.0 / 116.0)) * 3.0) * y;
}

↓

public static double code(double x, double y) {
	return (y * (3.0 * x)) + (y * -0.41379310344827586);
}

Python

def code(x, y):
	return ((x - (16.0 / 116.0)) * 3.0) * y

↓

def code(x, y):
	return (y * (3.0 * x)) + (y * -0.41379310344827586)

Julia

function code(x, y)
	return Float64(Float64(Float64(x - Float64(16.0 / 116.0)) * 3.0) * y)
end

↓

function code(x, y)
	return Float64(Float64(y * Float64(3.0 * x)) + Float64(y * -0.41379310344827586))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - (16.0 / 116.0)) * 3.0) * y;
end

↓

function tmp = code(x, y)
	tmp = (y * (3.0 * x)) + (y * -0.41379310344827586);
end

Wolfram

code[x_, y_] := N[(N[(N[(x - N[(16.0 / 116.0), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision] * y), $MachinePrecision]

↓

code[x_, y_] := N[(N[(y * N[(3.0 * x), $MachinePrecision]), $MachinePrecision] + N[(y * -0.41379310344827586), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus x

Bits error versus y

Target

Original	0.2
Target	0.2
Herbie	0.2

\[y \cdot \left(x \cdot 3 - 0.41379310344827586\right) \]

Derivation

1. Initial program 0.2

   \[\left(\left(x - \frac{16}{116}\right) \cdot 3\right) \cdot y \]

2. Taylor expanded in y around inf 0.3

   \[\leadsto \color{blue}{3 \cdot \left(y \cdot \left(x - 0.13793103448275862\right)\right)} \]

3. Applied add-cube-cbrt_binary64 0.3

   \[\leadsto \color{blue}{\left(\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \sqrt[3]{3}\right)} \cdot \left(y \cdot \left(x - 0.13793103448275862\right)\right) \]

4. Applied associate-*l*_binary64 0.5

   \[\leadsto \color{blue}{\left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{3} \cdot \left(y \cdot \left(x - 0.13793103448275862\right)\right)\right)} \]

5. Applied sub-neg_binary64 0.5

   \[\leadsto \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{3} \cdot \left(y \cdot \color{blue}{\left(x + \left(-0.13793103448275862\right)\right)}\right)\right) \]

6. Applied distribute-rgt-in_binary64 0.5

   \[\leadsto \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{3} \cdot \color{blue}{\left(x \cdot y + \left(-0.13793103448275862\right) \cdot y\right)}\right) \]

7. Applied distribute-rgt-in_binary64 0.5

   \[\leadsto \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \cdot \color{blue}{\left(\left(x \cdot y\right) \cdot \sqrt[3]{3} + \left(\left(-0.13793103448275862\right) \cdot y\right) \cdot \sqrt[3]{3}\right)} \]

8. Applied distribute-rgt-in_binary64 0.5

   \[\leadsto \color{blue}{\left(\left(x \cdot y\right) \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) + \left(\left(\left(-0.13793103448275862\right) \cdot y\right) \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right)} \]

9. Simplified 0.5

   \[\leadsto \color{blue}{y \cdot \left(3 \cdot x\right)} + \left(\left(\left(-0.13793103448275862\right) \cdot y\right) \cdot \sqrt[3]{3}\right) \cdot \left(\sqrt[3]{3} \cdot \sqrt[3]{3}\right) \]

10. Simplified 0.2

    \[\leadsto y \cdot \left(3 \cdot x\right) + \color{blue}{y \cdot -0.41379310344827586} \]

11. Final simplification 0.2

    \[\leadsto y \cdot \left(3 \cdot x\right) + y \cdot -0.41379310344827586 \]

Reproduce

herbie shell --seed 2022131 
(FPCore (x y)
  :name "Data.Colour.CIE:cieLAB from colour-2.3.3, A"
  :precision binary64

  :herbie-target
  (* y (- (* x 3.0) 0.41379310344827586))

  (* (* (- x (/ 16.0 116.0)) 3.0) y))